Recursive block sequence defined by greatest prime factors

[Ali Sada]

Hi everyone,

Hope all is well. I would really appreciate it if you could tell me whether this sequence is suitable for the OEIS. 

"Start with 1,2. If p is the greatest prime factor of the last term of the sequence, append, in increasing order, the p smallest positive multiples of p that have not previously appeared."

 

1,2,4,6,3,9,12,15,18,21,7,14,28,35,42,49,56,63,70,77,84,91,98,105,112,119,126,133,140,147,154, 11,22,33,44,55,66,88,99,110,121,132,143,165,176,187,198,209,220,231,242,253,264,275,286, 

297,308,319,330,341,352,363,374,385,396,407,418,429,440,451,462,473,484,495,506,23

Question: Do primes appear in this sequence in increasing order?

Best,

Ali

[Robert Israel]

Looks like an interesting sequence. I think it should be presented as a triangle

1,2

4, 6

3,9,12,

15,18,21

7, 14, 28, 35, 42, 49, 56

etc

so if p is the greatest prime factor of the last term in a row, the next row consists of the first p positive multiples of p that have not previously appeared.

I don't know the answer to your question, but the first 16 primes that appear (in rows 1 to 54) are indeed in increasing order:

2, 3, 7, 11, 23, 71, 107, 179, 181, 547, 821, 5749, 6899, 41399, 82799,
124199.

Cheers,

Robert

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[Ali Sada]

Thank you very much Robert. I really appreciate your response, as always. The draft sequence is https://oeis.org/draft/A398261.

Best,

Ali

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[Geoffrey Caveney]

Those primes that appear in this sequence must occur in increasing order. Consider each prime in the sequence. It occurs because it was the largest prime factor in the preceding term. The preceding term must also have as a factor the previous prime that occurred in the sequence, as a result of the definition of the sequence. Since the newly occurring prime was the largest prime factor in the preceding term, it must be larger than the previous prime in the sequence.

On Fri, Jul 24, 2026 at 11:23 AM Robert Israel <isra...@gmail.com> wrote:

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[Žiga Pirc]

Some potential sequences related to this:
Lenght of each row: 2, 2, 3, 3, 7, 7, 7, 11, 11, 11, 11, 23, 23, 23, 71, 71, 71, 107, 107, 107, 107, 107, 179, 181, 181, 181, 547, 547, 547, 821, 821, 821, 821, 821, 821, 821, 5749, 5749, 5749, 5749, 5749, 5749, 6899, 6899, 6899, 6899, 6899, 6899, 41399, 41399, 82799, 82799, 82799, 124199, 124199, 124199, 124199, 124199, 124199, 186299, 186301, 186301, 372607, 372607, 372607, 372607, 372607, 372607, 372607, 372607, 1490429, 1490429, 1490429, 1490429, 1490429, 2484049, 2484049, 2484049, 2484049, 2484049, 2484049, 2484049, ...

Runs of lenght of rows: 2, 2, 3, 4, 3, 3, 5, 1, 3, 3, 7, ...

Divided by the lenght of each row (without first row): 

2, 3,
1, 3, 4,
5, 6, 7,
1, 2, 4, 5, 6, 7, 8,
9, 10, 11, 12, 13, 14, 15,
16, 17, 18, 19, 20, 21, 22,
1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12,
13, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24,
25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35,
36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46,
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25,
26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48,
49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, ...

This last one is questionable, but the first 2 are probably suitable enough?

Žiga 

petek, 24. julij 2026 ob 19:41:04 UTC+2 je oseba geoffre...@gmail.com napisala:

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[Ali Sada]

Thank you very much Geoffrey and Žiga for your responses. I really appreciate them.

Geoffrey, can you please add this as a comment on the sequence?

Žiga, I would love to add these sequences with your help.

Best,

Ali    

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[Žiga Pirc]

The lenght of rows is https://oeis.org/draft/A398297, for runs feel free to submit it yourself since I am on my 3 drafts limit. 

Žiga

sobota, 25. julij 2026 ob 04:59:27 UTC+2 je oseba ali....@gmail.com napisala:

[Geoffrey Caveney]

An interesting variant of Ali Sada's sequence (https://oeis.org/draft/A398261 with discussion at https://groups.google.com/g/seqfan/c/fhwkB-NWSCI/m/F6XK__kVAwAJ) is to modify the definition in this way:

If p is the greatest prime factor and q is the smallest prime factor of the last term in a row, the next row consists of the first q positive multiples of p that have not previously appeared.

This results in more primes occurring in the sequence; an interesting property is the types of primes that occur more and less frequently in the sequence.

The variant sequence begins:

1, 2,

4, 6,

3, 9,

12, 15, 18,

21, 24,

27, 30,

5, 10,

20, 25,

35, 40, 45, 50, 55,

11, ...

If my preliminary computations and analysis are correct, the first primes that occur in this sequence appear to be 2, 3, 5, 11, 13, 17, 19, 23, 29, 47, 53, 59, 107, 109, 113, 127, 227, 229, 233, 239, 467, ...

The striking feature of this set of primes is the preponderance of pairs of Sophie Germain and safe primes (a pair such that p and 2p+1 are both prime). Among the first 21 primes in the sequence listed above, 16 of them are either Sophie Germain or safe primes (see A005384 and A005385).

This tendency occurs due to a particular typical phenomenon in the sequence, which can be illustrated by examining the rows of large multiples of 29. After the row ending in 29*30, a long sequence of rows of length 2 occurs, which is highly typical of this sequence. The following rows end in 29*32, 29*34, etc. But after 29*44, the next row is different because 29*46 already occurred previously in the sequence as 23*58, the term which then introduced 29 itself into the sequence. So after 29*44, the next row is 29*45, 29*47. Because the number 47 = 23*2 + 1 is prime, the following row comprises multiples of 47. This is the mechanism by which the Sophie Germain primes 23, 53, 113, 233 lead to the appearance of the corresponding safe primes 47, 107, 227, 467 in the sequence, with only one other prime occurring in between them.

If this typical phenomenon continues to occur in the sequence without other rare phenomena intervening, then the following primes occurring in the sequence may be 479, 487, 491 (SG), 499, 983 (safe), 991, 997, 1009, 1013 (SG), 1019, 2027 (safe), 2029, 2039 (SG), 2053, 4079 (safe), followed by a long sequence of 13 primes until the next Sophie Germain prime 4211, followed by 4217, 8423 (safe), ....

As Sophie Germain and safe primes become less frequent among larger numbers -- they are conjectured to be about as frequent as twin primes -- their preponderance among the primes in this sequence would naturally be expected to decrease. It is an unknown open question whether there are infinitely many Sophie Germain and safe primes, but it seems to be considered very likely that infinitely many of them exist.

Geoffrey

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[Ali Sada]

TBH, this seems like a more interesting sequence! 

Best,

Ali

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